Assessing risk In the financial markets with the extreme value theory : are extreme events more prevalent In the Bucharest stock exchange?

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AMSTERDAM BUSINESS SCHOOL

MSC BUSINESS ECONOMICS, FINANCE TRACK

MASTER THESIS

Assessing Risk In The Financial Markets With

The Extreme Value Theory

Are Extreme Events More Prevalent In The Bucharest Stock Exchange?

Supervisor: Dr. Philippe Versijp

Student: Gabriel-Nicolae Șerdin

January

2015

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Statement of Originality

This document is written by Student Gabriel-Nicolae Șerdin who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This paper wishes to analyze and quantify the extreme risk on two different stock markets, the London Stock Exchange and the Bucharest Stock Exchange respectively, analysis which is centered on the two main stock indexes - the FTSE 100 Index and the BET Index - which serve as benchmarks. The assessment of extreme risk is done using Extreme Value Theory, namely the Peak over Threshold method – this modeling being subsequently compared to classical market risk measurements.

The focus in the following chapters will be to quantify the efficiency of the EVT compared to that of the classical approaches as well as providing an overview of the expression of market risk on the aforementioned exchanges.

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I. Introduction

Financial risk management has been an ever disputed topic amongst theoreticians and practitioners alike. Historically speaking, various economic, statistical and mathematical methods have been deployed in risk assessment, many of them being quantitative measurements for financial risk managers.

Curiously enough, the relation between the financial world and its view on risk is paradoxical. The most eloquent examples are the major financial crises. The last financial crisis from 2007-2008 underlined the fragility of the banking systems at an international level. On the other hand, there is pressure to maintain or increase profitability, which results in a higher risk appetite, despite the risks. One of most puzzling and yet dangerous risk phenomenon that can occur on financial markets is given by extreme risk concurrencies, the like of “black swans” - as described in Taleb (2010).

Therefore, several quantitative techniques to measure risk have been proposed at an international level, as those made by the Basel Committee. “Revisions to the Basel II market risk framework” (2011) makes note of a demand to more thoroughly assess financial risk by banks, especially by those who perform transactions on financial markets. Economic banking capital has indeed been underestimated, compared to minimum capital requirements. This leads to an even more extensive analysis of the market risk to which financial institutions expose themselves.

Classical market risk estimation methods are given by Risk and conditional Value-at-Risk measurements. One issue concerning this matter is given by the Gaussian assumption of the distribution of financial asset returns. As statistical reality has shown - Mandelbrot(1963), Fama (1965) - these distributions present fat-tails, thus adaptations in this direction should yield more accurate risk estimates.

Extreme value theory, which was previously used in the fields of bio-technology, the study of climate conditions and extreme geographical events, as well as in engineering, has achieved an adequate adaptation within the field of finance as well.

From this point of view, the research sets out to test the following hypothesis: is the Extreme Value Theory a more suitable method of estimating the Value-at-Risk than that based on the Gaussian approach?

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6 This research goal is not only of academic interest, but of a practical one as well. The need for more robust risk assessment is of major interest to financial institutions when pursuing their objectives related to capital requirements and risk exposure.

To this extent, in order to assess how the Extreme Value Theory (henceforth, called EVT) can be implemented, two financial stock indexes have been chosen for modeling: the FTSE 100 Index from the London Stock Exchange and the BET Index, from the Bucharest Stock Exchange.

Based on the Value-at-Risk estimates that will be obtained, an answer to the following question can be given: are extreme events more prevalent in the Bucharest Stock Exchange than in the London Stock Exchange? Furthermore, the research can be extended in order to determine if the model is suitable for both exchanges? The aforementioned extreme events are defined as changes in the underlying index returns greater than two standard deviations.

This analysis can set the base for further investigation because even though research on markets in different stages – developed versus emerging, or in transition - has been previously performed, to our knowledge, there is none which encompasses the specifics of this setup.

The paper is structured as follows. The next section covers the main literature that has been written on the topic of the Extreme Value Theory, in the context of financial market risk. The third section revises the main methods in pursuing EVT, as well as those utilized in measuring market risk. Within data and descriptive statistics section, the financial time series are statistically modeled and the main econometric GARCH model is fitted. The results encompass the main findings of the EVT-based method used, as well as the statistical measures for Value-at-Risk and conditional Value-at-Risk. Robustness checks are also done to ensure the goodness of fit of the method deployed. The last part is given by the conclusions, where both the advantages and limitations of the EVT method are presented. Future directions regarding multivariate dynamic risk measurements are indicated.

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II.

Literature review

In order to test asset pricing hypotheses, assumptions on the evolution of the entity under analysis need to be made. The same assumptions are also required when optimizing portfolios, computing efficient frontiers, valuing instruments, performing risk management and building hedging strategies.

Throughout time, various approaches on distributions behavior have been used in the field of research. In studying portfolio selection and deriving the CAPM model, Markowitz (1952) and Sharpe (1964) adopt an assumption based on normality for asset returns. Later, Black and Scholes (1973) and Merton (1973) price options by assuming a Brownian motion for the price of the underlying asset, hence linking the returns behavior to a Gaussian distribution. The latter has become an epitome in finance, its use being carried on to this day in the financial system.

Longin (2005) proposes EVT as a mean to distinguish between distributions of returns, given the tendency to depart from normality. Data characterized by high kurtosis has a leptokurtic distribution. A fat-tailed empirical distribution implies that there are more extreme observations than predicted by the normal model, hence a higher volatility.

For fat-tailed distributions to be modeled, many papers suggest the log-normal distribution, the generalized error distribution and mixtures of normal distributions.

Despite the excess kurtosis, given that some of these distributions decay exponentially, they are to be considered thin-tailed. Hence, even though they could fit the empirical distributions, this would happen only up to a certain point - afterwards deteriorating significantly. The point of deterioration is found in the proximity of the extremes.

In his study of cotton prices, Mandelbrot (1963) suggests that the Paretian distributions fitted the data better than the Gaussian distribution did. This belief was extended to stock prices in 1965 by Fama. A volatility clustering effect is observed by Mandelbrot (1963), in that large changes are followed by other large changes while small ones are followed by other which are similar in size. This behaviour of the volatility is easily replicated by the ARCH processes of Engle (1982). Praetz (1972) and Press (1967) proposed mixed distribution models - e.g. Gaussian and unconditional t-Student. It is observed that such mixed distribution models manifest the heterogeneity specific to random phenomena and can be used to account for extreme price movements such as stock market crashes or irregular movements like the "day effect".

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8 Gençay (2004) suggests avoiding an attempt to fit a single distribution to the entire sample and rather investigate only the tails of the return distributions given that only those are important for extreme values. One tool to study the tail behavior of fat-tailed distributions is the EVT whose adaptability in modeling the tails is of great use when extrapolating the probabilities of the quantiles which lie outside the sample to be observed.

Extreme Value Theory is a well-established branch in probability theory with applications in numerous areas, from natural science - meteorology, oceanography, hydrology, pollution studies - to structural engineering or matters of daily life - like highway traffic control, as described in Galambos et al. (1994).

One example of such an application of EVT can be the problem of determining the right height of a dam in order to prevent potential flooding of low landscape regions - like many areas in the Netherlands - in time of extreme rainfall.

As Embrechts et al. (1999) suggests, EVT has extended its use in financial risk management due to its ability to fit extreme quantiles better than the conventional approaches when analyzing heavy-tailed data.

Referring to its previous uses, one can compare the need to determine the correct height of the dam to that of determining the minimum capital requirements in order to absorb the losses from financial shocks.

Danielsson (2011) points out some of the appealing facts about EVT like it being able to fit the extremes in one of three distributions classes - Gumbell, Weibull, Fréchet -, of which only one is fat tailed - the Fréchet -.

As presented in Gençay (2004), EVT is an adequate framework for the separate treatment of the tails of a distribution. This constitutes a strongpoint when compared to other models like the t-distributions, normal distributions or even ARCH-like distributions because it allows for asymmetry - a characteristic of financial return series.

McNeil (1999) assesses the role of extreme value theory in risk management as a method to model and measure extreme risks. The Peak-over-Threshold method is implemented in a stochastic volatility framework in order to provide estimates of the Value-at-Risk and conditional Value-at-Risk – known also as Expected Shortfall. McNeil's analysis points out that the methods based on the assumptions of normal distributions tend to underestimate tail risks whilst methods based on historical simulations provide imprecise estimates of tail risk.

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9 On the topic of EVT, the paper concludes that it is a method that carries the proper tools to enhance current risk management systems.

McNeil and Frey (2000) compare conditional EVT - the residuals are extracted from a structure where the stochastic volatility follows a GARCH-type model with GARCH errors, fitted to historical data by pseudo maximum likelihood - with other conditional approaches which assume normally distributed or t-distributed methods and also with static EVT. The results of the analysis undertaken confirm that the conditional normal estimates produce violations more often than the conditional EVT estimates. Moreover, the static EVT estimates produce violations more often in times of high volatility, being unable to adapt fast enough to the change in volatility. Thus, according to McNeil and Frey, conditional EVT is generally the best performing method to estimate the Value-at-Risk for the 95th percentile and above. Furthermore, the conditional normally distributed approach is incapable of estimating the Expected shortfall at the 95th percentile level.

Dimitrakopoulos et al. (2010) compares the forecasting ability of a wide array of VaR models, including the EVT based ones. This extensive research tackles with the issue of VaR model selection from multiple perspectives. First of all, the data under analysis comprises normal periods as well as crisis and post-crisis periods. Second, both emerging markets and developed ones are taken into consideration.

The results from Dimitrakopoulos et al. (2010) suggest that the PoT is amongst the most accurate options available when trying to quantify the market risk based on the VaR. Moreover, it is found that the most efficient VaR models are the same for developed markets as well as emerging markets, despite the differences between them.

Diebold et al. (2000) provides a different approach to the topic as it sets up to exemplify some of the pitfalls specific to the EVT methodology and how to avoid them. The extended list of shortcomings can include: heavy-tailed innovations, fat-tailed distributions, long memory and self-similarity.

The same paper warns on the drawbacks of having to make use of measures whose estimation doesn't follow a specific procedure. Such measures can thoroughly impact on the statistical significance of the entire research. In this sense, caution needs to be taken when choosing the levels of the threshold used in the Peak-over-Threshold implementation of the EVT as there is no said rule for this process. The chosen threshold level impacts in more than one way as lowering it would reduce the variance within the newly obtained data but at the same time this

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10 would increase the chance of contamination with non-extreme values. This issue will be further discussed and tackled in the following sections of this paper.

Furthermore, Diebold touches on the heteroskedastic nature of the returns of financial assets and the fact that EVT-related literature generally assumes the variables to be independent and identically distributed. Given that the same returns generally are subject to volatility clustering, generalizations to dependent data are not much of an option since they mostly stand in need of "anti-clustering" conditions on the extremes. Two means of improving estimations in such situations are presented.

On the one hand, using the block-maxima based approach would reduce the dependence in data values. However, in doing so new issues arise as that of determining a well-suited periodicity of said blocks of data. Such practice could account for downturns like seasonality, or year-end effect.

Otherwise, the tail of the conditional distribution can be estimated instead of that of the unconditional one. This is done by fitting a conditional volatility model, standardizing the data by the estimated standard deviation followed by an estimation of the tail index based on the standardized data. Hence, a correct pattern of the volatility is taken into account and the standardized residuals will be approximately iid, thus creating an adequate framework for the implementation of EVT.

All the above issues will be taken into consideration in the following sections and the suggestions for their avoidance will be implemented in the pursued methodology.

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III. Methodology and hypothesis

1. Hypothesis

The hypothesis to be tested in the following sections are:

 the Extreme Value Theory is a more suitable method of estimating the Value-at-Risk than that based on the Gaussian approach;

 extreme events are more prevalent in the Bucharest Stock Exchange than in the London Stock Exchange.

In order to test the first hypothesis, the two index series - the FTSE 100 Index and the BET Index - will be analyzed using the Box-Jenkins method. This will ensure the best fit of each time-series to an autoregressive moving average model.

A standard approach to the Box-Jenkins methodology comprises the following: first, the adequate model is selected by identifying the autoregressive and moving average components to be used. Afterwards, a parameter estimation is performed as to determine the coefficients that best fit the selected ARMA model. The last step of the methodology involves testing the quality of the model by ensuring its conformation to the characteristics of a stationary univariate process.

The Gaussian VaR calculation does not require any further data processing and analysis. Obtaining the VaR through the EVT is a more laborious process as this will require a separate data processing of the upper and lower tails of the distributions of the data under analysis. The Extreme Values can be singled out in more than one way. First, one could chose a specific timeframe and then split the distribution of returns in multiple periods according to this criteria. A new distribution is constructed based on the maximum (minimum) observed value of each period. This method is the earliest to be used and is known as the "Block Maxima-Minima" (BMM).

A more modern approach would be the "Peak-over-threshold" (PoT). Its construction is again based on the return distribution. The values representing the extremes are those that exceed a certain threshold - noted as .

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12 Both methods exhibit certain drawbacks and require a certain degree of finesse when deciding upon their specificities. The implementation of the EVT in this research will be carried out through the means of the Peak-over-threshold method.

The per se testing of the first hypothesis is done by comparing the VaRs obtained through the normal Gaussian and EVT methodologies with their out-of-sample correspondents - i.e. the 1 day VaR will be compared to the next day observed out-of sample value, whilst the 10 day VaR will be compared to the value observed 10 days after the last in-sample value. Whichever method yields out the more precise approximations of the aforementioned out-of-sample values will be considered the most suitable for estimating the VaR under the current scenario. This pattern of analysis will be extended to a backtest in order to ensure a measurement of the effectiveness of the chosen methodology.

The testing of the second hypothesis will be performed using the aforementioned VaR estimates as well as other indicators like the average return or the average return exceeding two standard deviations. The latter will serve as means of confirmation of the information extracted from the VaR estimates.

2. The Generalized Pareto Distribution

The goal of the Peak-over-Threshold method is to estimate a distribution function comprising the values of which are greater than the threshold . is defined as follows:

where is a random variable, is the predefined threshold, constitute the extremes and is the right endpoint of . can also be expressed:

Given that most of the values are lower than , the estimation of the conditional excess distribution function can be attained with the help of the Pickands-Balkema-De Haan theorem, stating the following:

For a large class of underlying distribution functions the conditional excess distribution function , for large, is well approximated by

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13 where for , is the Generalized Pareto Distribution (GPD).

is then constructed to equal the sum of and the GPD can be depicted as _{, hence a function of .}

is called the tail index or shape parameter. Whilst keeping the scaling parameter equal to one, the GPD can be modeled through this shape parameter. Nevertheless, modeling a fat tailed distribution requires to be positive.

The GPD embeds a number of other distributions. When , it takes the form of the ordinary Pareto distribution. This particular case is the most relevant for financial time series analysis since it is a heavy-tailed one.

For instance, when , the GPD converges to the exponential distribution, whilst for it is known as a Pareto II type distribution.

The GPD has an infinite variance for and when it has an infinite fourth moment. According to Dacorogna et al. (2001) the estimates of are usually less than 0.5 for the security returns or high frequency foreign exchange returns, thus implying that the returns have finite variance.

The Maximum Likelihood Estimation shall be used to compute the estimates of the parameters of the GPD.

Hosking and Wallis (1987) present evidence that for >0.5, the maximum likelihood regularity conditions are fulfilled and the maximum likelihood estimates are asymptotically normally distributed. Hence, the approximate standard errors for the estimator of can be obtained through maximum likelihood estimation.

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14 Assuming a sample , the log-likelihood function for the GPD equivalent to the logarithm of the joint density of the observations

3. Measures of Extreme Risk Estimation

The chosen measures of extreme risk - i.e. the Value-at-Risk (VaR) and Expected Shortfall (ES) - are expressed in terms of the distribution of returns . The VaR is the -th quantile of the distribution

where and is the inverse of .

The Expected Shortfall - the expected loss size when the VaR is exceeded - can be written as

Bearing in mind the aforementioned and substituting, can be rewritten as

This can be reduced to

For a given probability , the previous equation yields

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where is the mean of the excess distribution over the threshold .

Given the previous stated, the mean excess function for the GPD when would be

and hence the expected shortfall can be rewritten as

VaRp confidence intervals based on the log-likelihood can be calculated by reparametrizing the GPD as a function of and VaRp

The corresponding probability density function is

4. Backtest

Actual implementation of EVT doesn't carry much difficulty and yields excellent estimates where EVT holds. Unfortunately, EVT is improperly utilized in numerous occasions, as it should only be applied in the tail region - any attempt to extend towards the center would impact on its accuracy.

Choosing the right threshold can be tricky, in that it has to meet the requirements of the GPD theorem of Pickands. But in the quest to meet those requirements, one has to consider that

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16 picking a higher threshold would highly impact on the number of observations remaining available for estimating the parameters of the tail distribution function.

The purpose of VaR models is to maximize the degree in which the model confirms to the empirical distribution for a given time horizon and confidence level. In order to assess the performance of the EVT approach, in the results section a comparison of the out-of-sample estimations of VaR and ES will be carried over for both index series.

To ensure a measurement of the effectiveness of the chosen methodology, a backtest is to be applied for each individual return series. Backtesting is a statistical procedure where actual profits and losses are systematically compared to corresponding VaR estimates.

There are multiple ways of performing the backtest. The one used in this paper is constructed as follows.

Fixed data samples of 1000 observations are extracted from within the return series. The VaR and ES are estimated for the 1 day and 10 days horizon through the standard Gaussian method as well as the EVT Peak-over-Threshold approach.

The backtest continues with a comparison. The estimates of the VaR and ES are compared to the "next day return" - i.e. the first observation following the 1000 observations used when estimating the VaR and ES.

The 1 day VaR backtest meter has an initial value of 0. It will be incremented by 1 each time the EVT produces a more accurate VaR estimate than the Gaussian method - i.e. the difference between the VaR estimate and the next day return is the smallest - and it will decline by 1 otherwise. Following the same rules, backtest meters are constructed for the 1 day ES and 10 days VaR and ES estimates.

A positive backtest meter is an indicator of the fact that the EVT produced better estimates of the VaR - or ES - for more than half of the computations.

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IV. Data and descriptive statistics

The empirical analysis will be carried out on the evolution of two market indexes: the FTSE 100 Index of the London Stock Exchange (LSE) and the BET Index – of the Bucharest Stock Exchange (BVB).

The FTSE 100 Index has been chosen because it is a mirror of evolved investment markets, where both trade volumes and share turnover are high, allowing for efficient investing.

On the other hand, the Bucharest Stock Exchange is a young market in its current form, although its roots can be traced up to the late 19th_century.

Hence, the analysis of the BET Index will provide more insights on the risk to which an investor exposes himself when trading in this market and furthermore, what the probability of facing extreme events is. Moreover, such an analysis has further use in testing the impact of exogenous shocks on a market portfolio, pursuing the testing of the asymmetry and dependency between the two indexes, or the BET Index and another representative index for which the comparison can be of use.

The data on both indexes has been exported from Bloomberg. The testing sample contains the daily closing values of the indexes between the years 1998 and 2013. Data is extracted in MS Excel and all the calculations related to EVT will be carried out in Matlab R2012b. For certain statistical testing, Eviews 7 will be preferred.

Figure 4.1: The plot of the daily quoted close price of

the FTSE 100 Index. Figure 4.2: The plot of the daily quoted close price of the BET Index.

Figures 4.1 and 4.2 plot the daily quoted close price for both indexes. The evolution of the two indexes bears limited resemblance, mainly due to the fact that the two markets were in different stages of their life cycle throughout the analyzed period.

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18 For example, in the late ‘90s – early ‘00s Romania was just beginning its process of transition to an open economy therefore the dot-com burst didn’t affect the BET Index growth whilst the FTSE 100 Index manifested a steep drop.

On the other hand, by the time of the financial crisis of 2007-08, Romania’s economy gained exposure to foreign markets, bearing a strong dependency on foreign investments. Moreover, throughout the late ‘90s the banking system in Romania went through a massive privatization process. Hence, by the time of the crisis, it consisted mostly of subsidiaries of the greater foreign banks. Therefore, it was affected in the same way, if not maybe more, by this event.

The FTSE 100 Index data amounts to 4309 observations. The average value of the index for the selected period adds up to 5485.75, with a minimum of 3287 and a maximum of 6930.20 and its standard deviation being 799.13.

The BET Index data amounts to 3998 observations. The average value of the index for the selected period adds up to 3839.98, with a minimum of 281.09 and a maximum of 10813.59, with a standard deviation of 2748.96.

Moving forward in testing the stationarity of the time series, the autocorrelation and partial autocorrelation functions are analyzed.

The autocorrelation functions are designed as follows: the "lag" - i.e. the time span between observations - is shown along the horizontal axis, while the autocorrelation is shown along the vertical axis. Two horizontal lines mark the .05 and -.05 levels. These lines indicate the bounds for statistical significance - the .05 threshold corresponding to a 95% confidence level.

Randomness is described by autocorrelation levels between lags not exceeding the +/-.05 mark. After scrutinizing the results of the autocorrelation function in figure 4.3, one would notice straightforward that there is a high degree of correlation between the lags in the case of the FTSE 100 Index series - given that for all of the first 20 lags the autocorrelation function exceeds the selected statistical significance level.

The same can be said for the BET Index series, whose autocorrelation function of the first 20 lags is plotted in figure 4.4.

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Figure 4.3: FTSE 100 Index sample Autocorrelation

function. Figure 4.4: BET Index sample Autocorrelation function

Moreover, for the partial autocorrelation functions in figures 4.5 and 4.6, only the first lag exceeds the .05 threshold. These are indicators of an AR (1) type process.

Figure 4.5: FTSE 100 Index sample Partial

Autocorrelation function. Figure 4.6: BET Index sample Partial Autocorrelation function.

The stationarity of the processes will be tested through the means of the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests.

t-Stat ADF Test t-Stat PP Test LM-Stat KPSS Test Trend and Intercept -2.00910 0.59560 -2.09679 0.54690 0.63723 Reject H0 Intercept -1.96224 0.30390 -2.05181 0.26470 0.70343 Reject H0

None 0.14090 0.72680 0.13501 0.72510 - -

Table 4.1: FTSE 100 Index stationarity test results

The results from the table 4.1 indicate that the FTSE 100 Index series, in level values, is non-stationary. The null hypothesis of the ADF and PP tests, iterating the presence of a unit-root, fail to be rejected. Moreover, the null hypothesis of the KPSS test is rejected, thus reconfirming the non-stationarity of the series.

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20 t-Stat ADF Test t-Stat PP Test LM-Stat KPSS Test Trend and Intercept -1.28558 0.89100 -1.37485 0.86830 1.01889 Reject H0 Intercept -1.02216 0.74750 -1.08071 0.72560 4.54594 Reject H0

None 0.31259 0.77600 0.22301 0.75100 - -

Table 4.2: BET Index stationarity test results

The same conclusion can be drawn from the results of the tests applied to the BET Index and described in table 4.2. Hence both financial time series taken into consideration are non-stationary processes, which is what one would expect.

After analyzing the plots in figures 4.5 and 4.6, it is observable that both indexes follow a "difference stationary" process, as many financial time series do.

In order to advance in the model identification process, the series first need to be stationarized. This is a prerequisite, as working with stationarized series enables the obtaining of meaningful sample statistics - like means, variances or correlations - that can be used as descriptors of future behaviour. Box and Jenkins suggest an approach which requires differencing in order to achieve stationarity.

The returns of the time series are to be computed as continuously compounded returns:

The returns of the FTSE 100 Index and BET Index sample data can be observed in figures 4.7 and 4.8.

Figure 4.7: FTSE 100 Index logged returns Figure 4.8: BET Index logged returns

As it can be seen, at return level, the two series manifest strong similarities: the maximum return attained by the FTSE 100 Index over the analyzed period amounted to 0.0938, while that of the BET Index was 0.1056. The minimum return also suggests a high degree of resemblance

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21 between the returns of the two indexes, with the FTSE 100 Index return going as low as -0.0926 while the BET Index return attained a minimum of -0.1312.

The similarities are also observable when referring to the average return level. Over the analyzed period, the FTSE 100 Index exhibited an average positive return of 0.0086 and a negative one of -0.0091 while the BET Index reached an average positive return of 0.0118 and a negative one of -0.0118.

However, when looking at the average returns exceeding two standard deviations, a slight tendency of the BET Index towards more extreme returns can be observed. In this sense, when the FTSE 100 Index return exceeds the two standard deviations level, it reaches on average 0.0361 for a positive shift and -0.0355 for a negative one. The BET Index return exceeding the two standard deviations level attains on average 0.0513 for a positive shift and -0.0549 for a negative one.

The new series have gone through the previously described tests. In both cases, the stationarity is confirmed by the results of the three tests.

As seen in tables 4.3 and 4.4 the null hypotheses of the ADF and PP tests are rejected while those of the KPSS test fail to be rejected.

t-Stat ADF Test t-Stat PP Test LM-Stat KPSS Test Trend and Intercept -30.85929 0.00000 -66.09269 0.00000 0.04766 Do Not Reject H0

Intercept -30.85952 0.00000 -66.09722 0.00010 0.07118 Do Not Reject H0

None -30.86022 0.00000 -66.10348 0.00010 - -

Table 4.3: Stationarity test results for the FTSE 100 Index returns series

t-Stat ADF Test t-Stat PP Test LM-Stat KPSS Test Trend and Intercept -53.27660 0.00000 -53.70080 0.00000 0.18250 Do Not Reject H0

Intercept -53.28120 0.00010 -53.70760 0.00010 0.21880 Do Not Reject H0

None -53.24510 0.00010 -53.77370 0.00010 - -

Table 4.4: Stationarity test results for the BET Index returns series

Knowing that the issue of stationarity has been addressed, the following step in the Box-Jenkins methodology is to identify the order of the autoregressive and moving average terms.

The analysis carried over from now on will be based on the newly obtained return series which, as seen, satisfy the stationarity conditions.

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22 Numerous approaches for identifying and exist, however, the main criteria for model selection amongst the models will be the Akaike Information Criterion (AIC) and the Bayesian information Criterion (BIC).

0 1 2 3 4 5 0 -5.909029 -5.911377 -5.918907 -5.921613 -5.922549 1 -5.908935 -5.913430 -5.913941 -5.921894 -5.921732 -5.922509 2 -5.910766 -5.914038 -5.913918 -5.922368 -5.921882 -5.924321 3 -5.917999 -5.921105 -5.922069 -5.921762 -5.921868 -5.924111 4 -5.920170 -5.920742 -5.921622 -5.922310 -5.922121 -5.923465 5 -5.922620 -5.923177 -5.924611 -5.924131 -5.924230 -5.923840

Table 4.5: Akaike Schwarz values for the p and q estimation test of the FTSE100 Index return series.

The role of the AIC is to estimate the information lost when a certain model is used to represent the process generating the data under analysis. The AIC is constructed as the difference between the number of parameters of the model and the maximized value of the likelihood function for the model.

Tables 4.5 and 4.6 display the values of the Akaike Information Criterion for each setup. The values which are not bolded correspond to setups whose coefficients are not statistically significant and will therefore be dropped. The valid combinations for the FTSE 100 Index return series are the , the and the . For the BET Index return series the , the and the are the setups for which the AR and MA terms are relevant.

0 1 2 3 4 5 0 -5.274709 -5.274682 -5.274183 -5.273844 -5.274417 1 -5.274948 -5.274496 -5.274163 -5.273738 -5.276801 -5.276251 2 -5.275058 -5.274616 -5.274235 -5.274754 -5.274839 -5.279996 3 -5.274462 -5.273986 -5.276866 -5.278462 -5.277115 -5.277651 4 -5.273964 -5.276712 -5.277467 -5.278395 -5.279766 -5.279469 5 -5.276232 -5.278000 -5.282174 -5.281913 -5.281417 -5.280940

Table 4.6: Akaike Schwarz values for the p and q estimation test of the BET Index return series.

When using the AIC as a means to select the best model within the available options, the preferred model should always be the one having the minimum AIC value. This is why the tested

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23 combinations did not exceed - even though an increase in the number of parameters could have enhanced the goodness of fit of the model, it would most definitely have had a negative impact on the AIC value. From tables 4.5 and 4.6 it can be seen that the best option - for both series - is the setup.

However, it can be seen from the squared autocorrelation function of the residuals in figures 4.9 and 4.10 that there seems to be a high degree of heteroskedasticity and serial correlation present among each of the innovations.

Figure 4.9: FTSE 100 Index Squared Residuals

Autocorrelation function Figure 4.10: BET Index Squared Residuals Autocorrelation function

In order to confirm this visual indicator, the analysis will proceed with the Ljung-Box test for serial correlation and with the ARCH test for heteroskedasticity. For further analysis and confirmation, the White and Breusch-Godfrey-Pagan tests will be deployed.

Interval H Probability t-Statistic t-Critical value

1% 1 0.00% 159.2911 37.5662

Table 4.7: Ljung-Box Test on the FTSE 100 Index return series

Interval H Probability t-Statistic t-Critical value

1% 1 0.00% 94.6769 37.5662

Table 4.8: Ljung-Box Test on the BET Index return series

The Ljung-Box test results in tables 4.7 and 4.8 confirm the serial correlation and present at the residual level of the returns series. This was not unexpected, as numerous financial time series exhibit such behaviour. In order to continue with the econometric analysis, the following step would be to build-up a GARCH model. The assumptions are that the model is part of the exponential GARCH models, but these assumptions will be relaxed later on, by incorporating the leverage effect, introduced by Glosten, Jagannathan and Runkle (1993).

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24 The first attempt will be to incorporate the FTSE 100 Index and BET Index return series to the E-GARCH class models defined as:

From tables 4.9 and 4.10 of the Appendix, the models that would provide the best fit are the ARMA(1;1) – GARCH(1;1;1) for the FTSE 100 Index return series and the AR(1) – E-GARCH(1;1;1) for the BET Index return series.

Testing revealed that the t-Student distribution provides the most adequate fit for the model's residuals of the FTSE 100 Index returns. The GARCH specification was needed in the first place in order to encompass the conditional heteroskedasticity and serial correlation present amongst the error terms and see if by using this specification, the above correlation and heteroskedasticity are explained.

With a p-Value of 0% the White test reveals that in the case of the FTSE 100 Index return series the heteroskedasticity has not been explained.

Based on the same line of interpretation, the BET Index returns are to be put under analysis based on the estimates in table 4.10 from the Appendix. As previously mentioned, the specification of the model for this return time series is AR(1) – E-GARCH(1;1;1).

As in the case of the model's residuals of FTSE 100 Index returns series, testing results suggested that the t-Student distribution is the most adequate fit for the model's residuals of the BET Index return series. The presence of serial correlation and heteroskedasticity is observed at innovations level. This is confirmed by the White test results in the Appendix, the interpretation of which remains the same as previously noted.

Even if the E-GARCH framework does an acceptable job when it comes to fitting the data under analysis to the model, due to its inability to explain the heteroskedasticity and serial correlation it will not be further considered.

Another option to be pursued is that of attempting to fit the FTSE 100 Index and BET Index return series to an ARMA-GJR framework, defined as below:

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25

The Glosten, Jagannathan and Runkle (1993) framework has proved to be an important improvement in equity time series estimation as it includes another element to the conditional volatility equation which tries to describe the volatility clustering phenomena present in most of the equity financial series. This extra term incorporates the leverage effect. In the equity markets, when prices drop the volatility increases more than when prices increase in the same proportion. A higher negative return leads to a higher volatility. From an economic perspective, this behaviour can be explained by the leverage effect - Debt to Capital. When the price of a stock falls, on the short term, the level of debt of the issuing company remains constant, therefore the leverage effect rises.

Given that one of the goals of this paper is to offer the means to quantify the Value-at-Risk that the return series face, if the GJR model represents a better fit than the E-GARCH one, then the error terms will have had the heteroskedasticity and serial correlation explained. Therefore, one would assume that the volatility clustering effect can be explained through the leverage effect. Moreover, the innovations that will be extracted in order to model extreme shocks would prove to be better candidates, as they would be independently and identically distributed.

Tables 4.11 and 4.12 in the Appendix contain the relevant information providing confirmation to the beliefs that the GJR is a better fit, both series being modeled in a ARMA(1;1)-GJR(1;1;1) setup.

Furthermore, the heteroskedasticity and serial correlation phenomena have been explained. The Ljung-Box statistic confirms the explaining of the serial correlation phenomenon and the ARCH statistic confirms that the heteroskedasticity has been fully addressed within this framework. The remaining of this section addresses the issue of fitting the BET Index returns series to the GJR framework, as it has been previously achieved for the FTSE 100 Index.

The heteroskedasticity and serial correlation in this specification of the BET Index have been explained, as confirmed by the Ljung-Box and ARCH tests.

The two return series have achieved the best fit in an ARMA(1;1)-GJR(1;1;1) framework. This setup has been chosen due to its performance in explaining the volatility clustering effect and eliminating the non-constant variance from the error terms.

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26 The latter will be extracted and modeled in order to assess the extreme exogenous shocks that can affect the returns time series. All further data analysis will be carried out on the innovations of the return series.

When scrutinizing the residuals series, a preservation of the previous patterns is again observable - i.e. a high degree of similarity between the two indexes, with a slight inclination of the BET Index towards more extreme values.

While the FTSE 100 Index minimum and maximum innovations are equal to -5.9852 and 3.6014 respectively, those of the FTSE 100 Index amount to -6.0077 and 6.1637 respectively. Average innovations exceeding the two standard deviation level indicate the same behaviour, with the FTSE 100 Index having an average positive innovation of 2.3466 and an average negative one of -2.5319, while for the BET Index they amount to 2.6391 and -2.6091 respectively.

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27

V. Results

As previously mentioned, once the models that best fit the return series of the indexes are established and after the innovations have been extracted - i.e. exogenous shocks, from an economic perspective -, the market risk will be quantified through the VaR under two approaches: the EVT Peak-over-Threshold and the Normal Gaussian based method. The latter has been chosen as a mean of comparison due to the widespread use of the Gaussian VaR in the measuring of market risk.

The motivation behind choosing the EVT Peak-over-Threshold method is therefore twofold: on the one hand, it is desired to investigate whether this method would provide a better understanding of extreme shocks in the equity markets, while on the other hand, it follows quantitative research already present in the literature (Gilli (2006); McNeil et al. (2010); Malevergne et al.(2006)).

Figure 5.1: Mean Excess Function for the standardized

residuals of the FTSE 100 Index Returns Figure 5.2: Mean Excess Function for the standardized residuals of the BET Index Returns

The first step in implementing the EVT Peak-over-Threshold method is to select an appropriate threshold level – . The threshold is selected intuitively by plotting the mean excess function against the values of the innovations, approach consistent with Gilli (2006).

By analyzing figures 5.1 and 5.2, the threshold is pinned at a level where it is believed that the function is losing its linear shape. Even though a reasonably estimate of the location for is pinned this way, this step can be seen as a drawback of the methodology due to its reduced statistical accuracy. -4 -3 -2 -1 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 u

FTSE 100 Returns Standardized Residuals Mean Excess Function

-8 -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6 7 u

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28 For the FTSE 100 Index series, the threshold is located between 1.50 and 2 for both its lower and upper tail. For the BET Index series, the threshold is lower, located between 1 and 1.50 for both its lower and upper tail.

Equilibrium needs to be achieved when selecting the threshold value to be further used. This needs to be high enough in order to capture the extreme component of the analysis for both tails. However, choosing a level too high would translate into a loss of important data which is used in estimating subsequent parameters. Therefore, the parsimony principle is also applied in its choice. In addition, for consistency purposes, robustness checks of the distribution fit need to be performed each time a different threshold is chosen.

By incorporating the above thresholds, the remaining datasets will contain 303 and 262 observations for the FTSE 100 Index lower and upper tails respectively. As for the BET Index, they will contain 280 observations for the lower tail and 300 for the upper one. The thresholds correspond to certain returns associated with the -th innovation on a specific day from the sample data.

The values of the innovations, the level , the number of observations and the corresponding quantile from the distribution of the error terms for both indexes can be found in table 5.1.

FTSE 100 Index BET Index

Lower Tail Upper Tail Lower Tail Upper Tail

0.075 0.935 0.070 0.925 Observations 303 262 280 300 -0.05666 0.018571 0.025044 0.03466 Errors 0.054067 0.057848 0.060432 0.061504 0.417873 0.525276 0.635892 0.601013 Errors 0.032938 0.044453 0.054037 0.050686 Value 1.519993 1.614472 1.358483 1.320518 Innovation value 0.016262 0.017552 0.021214 0.020338

Table 5.1: Selection of the threshold level u for both tails of the FTSE 100 Index and BET Index

More detailed data is presented in the Appendix in tables 5.2 and 5.3 for the FTSE 100 Index and 5.4 and 5.5 for the BET Index. The same tables also depict the maximum likelihood estimates for and . These have been estimated using an in-house function built in Matlab.

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29 Figures 5.3 and 5.4 present the QQ-plots of the sampled data versus the normal distribution for the tail and the scale parameters estimates of the FTSE 100 Index. A QQ-plot is a better way to assess normality than a histogram, because non-normality shows up as points that do not approximately follow a straight line. It can be observed in the figures, both estimators are normally distributed.

This has been also confirmed by the results of the Jarque-Bera tests that failed to reject the null hypothesis that the tested samples come from normal distributions.

Figure 5.3: QQ-plots for the calculated estimators of the

lower tail of the FTSE 100 Index Figure 5.4: QQ-plots for the calculated estimators of the upper tail of the FTSE 100 Index

The QQ-plots of said parameter estimates of the BET Index can be observed in figures 5.5 and 5.6. Reconfirmation the normality is again achieved from the results of the Jarque-Bera tests.

Figure 5.5: QQ-plots for the calculated estimators of the

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30 The GPD is expected to be an optimal fit for the series of extremes. However in order to ensure full statistical relevance of its parameters estimates and confirm that the distribution is a GPD one, the Kolmogorov-Smirnov, the Anderson-Darling and the goodness-of-fit tests are to be performed. Detailed results – which sustain the choice made - and step-by-step description of the tests are presented in the robustness section of this paper.

The following step is to determine the VaR measure under the PoT approach, as presented in the methodology section of this paper. The VaR and ES outputs obtained through the PoT method are compared with those obtained through the normal Gaussian approach, in order to determine which methodology provides a better use of these measures of risk. The main results are presented in tables 5.6 and 5.7.

An important annotation needs to be made at this point in order to ease the understanding of the interpretation of the results that will follow. Since the data in the tails consists of the standardized residuals of the ARMA-GJR models used, the risk measures estimates depict changes relative to the innovations values and not to those of the index itself or its returns. Scrutinizing the VaR and ES estimates for the FTSE 100 Index obtained by implementing the EVT approach one would conclude that for a 1 day horizon there is a 99% probability that the loss generated by the change in the shocks will not exceed 231.59% of the previous day shock value for a long position on the index and if it would, the average level that it would attain will be 266.87% of the previous day shock value. For a short position, the aforementioned levels would be 261.40% and 316.81% respectively.

Under the Gaussian framework, the estimates for the same timeframe at a 99% probability would imply that the variation of the innovations would not exceed 230.66% of the previous day shock value and if it were to surpass this level, the average change would be of 264.57%.

The VaR and ES estimations are the same both for the lower and upper tail in the Gaussian context as the standard normal distribution is symmetric.

Considering the "T-th" observation as the last of the series, data will be imported for the T+1 observation of the FTSE 100 Index in order to determine the effective change in the standardized residuals. Based on the results of this computation it can be stated that for the 1 day time horizon the VaR and ES estimated through the EVT Peak-over-Threshold are the most effective ones. This result is in line with the assumption of the first hypothesis stating that the EVT is a more suitable method for estimating the VaR than the Gaussian approach.

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31 Moving on to the 10 day horizon, based on the computations estimates it can be affirmed that under the EVT assumption there is a 99% probability that the loss incurred by the change in the shocks will not exceed 403.68% of the shock value at day T for a long position on the index and if it would exceed that level, it would attain on average 429.73% of the shock value observed at day T. For a short position, the aforementioned levels are 477.53% and 537.03% respectively. The Gaussian based estimates for the same timeframe at a 99% confidence level would imply that the variation of the innovations would not exceed 733.98% of the shock value observed at day T and if it were to surpass this level, the average change would be of 841.20% of the shock value at day T.

EVT Gaussian Innovations Variation

Lower Tail Upper Tail

1 Day VaR 95% 1.6877 1.7517 1.6247 Effective 1 Day Change

99% 2.3159 2.614 2.3066 -2.7027

ES 95% 2.0742 2.2895 2.0428 Max. 1 Day Change

99% 2.6687 3.1681 2.6457 -2.7027

10 Days VaR 95% 2.0503 2.0483 5.1835 Effective 10 Days Change

99% 4.0368 4.7753 7.3398 2.0816

ES 95% 2.4174 2.5918 6.5056 Max. 10 Days Change

99% 4.2973 5.3703 8.4120 -3.0013

Table 5.6: VaR and ES estimates from the EVT and Gaussian approaches for the FTSE 100 Index. The standardized residuals change for a 1 day and 10 days period.

The VaR and ES estimates obtained through the EVT reconfirm the effectiveness of this methodology, as they are the ones to best predict the change to which the innovations are subject. This is again in line with the assumption of the first hypothesis tested in this paper. The same analysis is continued for the Romanian index. Similar interpretations apply: the VaR and ES estimates obtained by implementing the EVT approach for a 1 day horizon imply that with a 99% probability the loss generated by the change in the shocks will not exceed 262.70% of the previous day shock value for a long position on the index and if it would, the average level that it would attain would be 331.18% of the previous day shock value. For a short position, the aforementioned levels are 257.53% and 324.29% respectively.

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32 Under the Gaussian framework, the estimates for the same timeframe at a 99% probability imply that the variation of the innovations will not exceed 231.06% of the previous day shock value and if it were to surpass this level, the average change would be of 264.69%.

EVT Gaussian Innovations Variation

Lower Tail Upper Tail

1 Day VaR 95% 1.5738 1.5664 1.6342 Effective 1 Day Change

99% 2.6270 2.5753 2.3106 -6.3688

ES 95% 2.2316 2.1978 2.0489 Max. 1 Day Change

99% 3.3118 3.2429 2.6469 -6.3688

10 Days VaR 95% 2.0395 2.0980 5.1640 Effective 10 Days Change

99% 5.3699 5.2884 7.3028 1.7706

ES 95% 2.7092 2.7485 6.4754 Max. 10 Days Change

99% 6.1252 6.0535 8.3663 14.6325

Table 5.7: VaR and ES estimates from the EVT and Gaussian approaches for the BET Index. The standardized residuals change for a 1 day and 10 days period.

For the 10 Day horizon, the computations are marked by the same high differences between the Gaussian ones and the EVT ones. At the 99% probability level, the loss incurred by the change in the shocks will not exceed 536.99% of the last observed shock value for a long position on the index and if it would, the average level that it would attain would be 612.52% of the shock value at day T. For a short position, the aforementioned levels are 528.84% and 605.35% respectively. The Gaussian based estimates for the same timeframe at a 99% confidence level imply that the variation of the innovations will not exceed 516.40% of the shock value at day T, and if it were to surpass this level, the average change would be of 836.63%.

When comparing with the effective change that developed in the 10 days period, the VaR estimates obtained through the EVT PoT method provide a more accurate measure of risk. However, in the case of the ES indicator, the Gaussian based estimation yields a more proper feel of the risk.

As mentioned before, all the results presented throughout tables 5.6 and 5.7 and their interpretations above are relative to the values of the innovations. When translating into absolute values these changes would be equivalent to around 30 to 70 basis points or around 0.01% to 0.02% of the index value.

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33 corresponds to a decrease of 32 basis points or 0.005% of the previous day index value or 3.8% of the day on day return value.

Overall, the EVT estimates for the VaR applied to the residuals perform better than the Gaussian counterparts for the 1 Day horizon, at both the 95% and 99% confidence levels. If applied within the context of the econometric model, it is seen that the errors, which are treated as exogenous shocks, contain only little information as to the price at the next period.

The test results confirm that the Extreme Value Theory is a more suitable method of estimating the Value-at-Risk than that based on the Gaussian approach. These results are in line with the assumption of the first hypothesis for both the 1 day and 10 days timeframes. Moreover, the same behaviour is observed in the results of the analysis on both indexes.

The consistency of these results is tested through the means of the backtesting procedure, whose results are presented in table 5.8.

Results FTSE 100 Index Results BET Index

VaR Day 95% 1026 1613 99% 1026 1613 Ten Days 95% 1220 1075 99% 1344 1415 ES Day 95% -2548 -1955 99% -2487 -1967 Ten Days 95% 2352 2079 99% 602 582

Table 5.8: Backtesting VaR and ES results for FTSE 100 and BET

It can be observed from table 5.8 that the results of the backtest of the Value-at-Risk measure are satisfactory and can be interpreted thusly. Firstly, concerning the FTSE 100 Index, the EVT based solution tops that based on the Gaussian framework by 1026 observations out of a total sample of 3028 - for a 1 day horizon at either 95% or 99% confidence level. The results on the BET Index provide a similar degree of consistency, with even higher inclination towards the EVT approach given the lower number of total observations.

Moreover, the VaR backtest results - which again come as a confirmation of the first hypothesis - are in line with related literature, as McNeil (1999) where the conditional EVT turns out to be the best method for estimating VaR for a confidence level higher than 95% for a 1 day horizon.

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34 The ES estimates however, manifest different behaviour. While the 10 days risk measure estimates based on the EVT at a 95% confidence level top by around 2000 observations the Gaussian based ones, the 1 day horizon estimates support the Gaussian approach. Needless to say, the results follow the same pattern for both series. This challenges the results in tables 5.6 and 5.7 in the matter of the ES.

Overall, it can be concluded that the backtesting results are at least satisfactory and consistent with the identification of the models. The VaR measures that have been modeled and quantified present a high degree of robustness.

When putting a comparative perspective on the VaR estimates and backtest results it can be said that they are consistent with previously made observations, in that there is a strong similarity between the two series – being in line with related literature as Dimitrakopoulos et al. (2010). The differences between the VaR and ES estimated for the FTSE 100 Index and those of the BET Index for a 1 day timeframe revolve around 5 decimal points. Judging by the amplitude of the ES it can be said that the slight predilection of the BET towards more extreme values, observed at both return and standardized residuals levels, is again reconfirmed.

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35

VI. Robustness checks

In order to ensure a high degree of statistical relevance of the EVT results and receive confirmation that the GPD distribution is a proper fit, the Kolmogorov-Smirnov, the Anderson-Darling and the Chi-squared ( ) goodness-of-fit tests are to be performed.

The Kolmogorov – Smirnov test is perhaps one of the most known goodness-of-fit tests. This applies to continuous univariate distributions and it measures the distance between the empirical distribution function of the data set and a random generated GPD distribution with estimates and , as follows:

,

where is the empirical cumulative distribution function and is the GPD cumulative distribution function.

The test null hypothesis states that the analyzed distribution falls within the type of distribution against which it is tested. Individual testing has been performed for each tail series.

Lower tail Upper tail

Sample Size 303 262

t-Statistic 0.02957 0.0393

p-Value 0.94665 0.79799

Confidence level t-Critical t-Critical

90% 0.07026 0.07556

95% 0.07802 0.08390

99% 0.09358 0.10064

Table 6.1.: Kolmogorov-Smirnov test results for the FTSE 100 Index lower and upper tails

From Table 6.1 it can be seen that for each tail, the goodness-of-fit test fails to reject the null hypothesis that the chosen distribution is a good fit. In this setting, a t-Statistic value inferior to the t-Critical level of significance would indicate a failure to reject the null hypothesis. It can be therefore concluded that the extreme values of the FTSE 100 Index series are well fitted against the GPD distribution and that the chosen threshold is a good choice.

Plotting of the FTSE 100 Index sample cumulative density functions (CDFs) – for the upper and lower tails – against the Generalized Pareto CDF is available in figures 6.1 and 6.2 of the Appendix.

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36

Lower tail Upper tail

Sample Size 280 300

t-Statistic 0.03028 0.01911

p-Value 0.95267 0.99983

Confidence level t-Critical t-Critical

90% 0.07309 0.07061

95% 0.08116 0.07840

99% 0.09735 0.09405

Table 6.2.: Kolmogorov-Smirnov test results for the BET Index lower and upper tails

Table 6.2 exhibits the results of the same test for the BET Index lower and upper tails. The Kolmogorov-Smirnov test indicates the same results as for the FTSE 100 Index: the GPD distribution is a good fit for the extreme values of the BET Index lower and upper tail’s observations, as the t-Stat is inferior to the t-Critical level of confidence for the 90%, 95% and 99% confidence levels.

Plotting of the BET Index sample CDFs – upper and lower tails – against the Generalized Pareto CDF is available in figures 6.3 and 6.4 of the Appendix.

For further confirmation of the robustness of the fit, the Anderson-Darling test will be applied. This test is similar to the previous one as it computes the difference between the empirical cumulative distribution function given by the dataset and the hypothesized GPD distribution. The test belongs to the class of quadratic EDF statistics. The EDF quadratic statistic measures the distance between and :

where is a weight function. If this is equal to 1, then the test becomes the Cramer-von-Mises test. However, only the Anderson-Darling test needs to be computed, which is based on the following statistic:

The tests null hypothesis assumes that the distribution falls within the type of distribution against which it is tested.

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37

t-Statistic Lower tail 0.36235 t-Statistic Lower tail 0.36235

Upper tail 0.62833 Upper tail 0.62833

Confidence level t-Critical Confidence level t-Critical

90% 1.9286 90% 1.9286

95% 2.5018 95% 2.5018

99% 3.9074 99% 3.9074

Table 6.3: Anderson-Darling test results for the FTSE

100 Index lower and upper tails Table 6.4: Anderson-Darling test results for the BET Index lower and upper tails

Tables 6.3 and 6.4 contain information regarding the statistics for the extreme values of the indexes. For the FTSE 100 Index extremes, the null hypothesis is not rejected and infers that the observations are drawn from a GPD distribution both for the upper and lower tail. In the case of the BET Index extreme values, the observations also fit well the GPD distribution, as the null fails to be rejected in this case as well.

The last goodness of fit test is the (Chi-square) test, which is similar to the previous two, with two differences. Firstly, this test can also apply to discrete distributions, not only continuous ones. Secondly, the test computes the difference between the cumulative empirical distribution and a hypothesized one, but it makes use of binned data. This can be viewed as a restriction in the case for smaller data samples, but in the current study, a number of observations above 200 will suffice to test the statistic:

where represents the total number of bins, is the observed frequency for bin and is the expected frequency for the same bin.

The expected frequency is given by:

where is the cumulative distribution function for the distribution being tested, is the upper limit for class and is the lower limit for the same class, while is the sample size.

The hypotheses remain the same as for the previous two tests, i.e. states that the distribution falls within the type of distribution it is tested against.

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38

Lower tail t-Statistic 4.0481 Lower tail t-Statistic 10.689

p-Value 0.85276 p-Value 0.21995

Upper tail t-Statistic 3.6261 Upper tail t-Statistic 4.1584

p-Value 0.88919 p-Value 0.84255

Confidence level t-Critical Confidence level t-Critical

90% 13.362 90% 13.362

95% 15.507 95% 15.507

99% 20.090 99% 20.090

Table 6.5: Chi-Squared test results for the FTSE 100

Index data Table 6.6: Chi-Squared test results for the BET Index data

As it can be seen from tables 6.5 and 6.6, in the case of both data series, the extreme values fall within the class of desired distribution, i.e. the Generalized Pareto distribution.

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39

VII. Conclusions

The present paper aimed at quantifying extreme market risk for two stock indexes: LSE’s FTSE 100 Index and BVB’s BET Index, both of them being representative for the capital market under which they were found.

In the fourth section, the statistical processing of the data has been performed. Firstly, the need to assess the time series under a stationary process was obvious. Being in line with financial literature, neither of the series was stationary in level values.

The process of stationarizing the series has been put into place. This included passing into first difference, followed by the identification of the model within the autoregressive moving average framework and extracting the residuals afterwards. The presence of heteroskedasticity, serial correlation and normality was verified at an error term level. Given that the residuals exhibited such behavior, it was obvious that volatility varied in time with the returns. Furthermore, serial correlation was another indicator that the heteroskedasticity phenomenon needed to be properly addressed, under a conditional heteroskedasticity framework, namely GARCH.

The choice of said model was divided between two classical financial time series models: the E-GARCH framework and the GJR one. Both of these models were however identified in order to clearly assess the differences between them and to see which fitted better the dynamics of both the conditional mean and volatility equations, within an ARMA-GARCH framework. As it was seen, the best model to fit the FTSE100 Index return series was the ARMA(1;1)-GJR(1;1;1). The same setup has been used for the BET Index return series. The tests performed confirmed the presence of homoskedasticity and absence of serial correlation. Following related literature on the topic, the residuals were extracted and standardized by the estimated standard deviation. From that point, the analysis that followed was developed using the newly constructed residual series.

The next step of the analysis consisted of choosing the right threshold , so as to proceed with the Extreme Value Theory methodology. A multi-stage process has been implemented in order to fulfill this task.

First, the mean excess function graph was drawn, by plotting the excess values above a certain level against the values of the series themselves. The choice of the threshold value was made taking into consideration the following: the number of values that exceeded the threshold should have been sufficient, in order to model the distribution, but not so many as to lose the extreme